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Abstract: We consider definably complete and Baire expansions of ordered fields: everydefinable subset of the domain of the structure has a supremum and the domaincan not be written as the union of a definable increasing family of nowheredense sets. Every expansion of the real field is definably complete and Baire.So is every o-minimal expansion of a field. However, unlike the o-minimal case,the structures considered form an elementary class. In this context we prove aversion of Kuratowski-Ulam-s Theorem and some restricted version of Sard-sLemma. We use the above results to prove the following version of Wilkie-sTheorem of the Complement: given a definably complete Baire expansion K of anordered field with a family of smooth functions, if there are uniform bounds onthe number of definably connected components of quantifier free definable sets,then K is o-minimal. We further generalize the above result, along the line ofSpeissegger-s theorem, and prove the o-minimality of the relative Pfaffianclosure of an o-minimal structure inside a definably complete Baire structure.

Autor: Antongiulio Fornasiero, Tamara Servi


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